Solve for every a the equationsystem:

$\displaystyle

x+y+az=3

$

$\displaystyle

2x-y+3x=4

$

$\displaystyle

x+ay+z=3

$

I put the system in a matrix and did elimination until:

1 2 a 3

0 -3 (3-2a) -2

0 (a-1) (1-a) 0

which gives me a = (y-z)/(y-z)=1

Is this correct, I'm not very used to this kind of exercises with unknown coefficients so I'm rather unsure at this. Am I going in the right direction? If not I'd be grateful for any help!